# Equivalence relation on V

Trying to prove equivalence relation on $$V$$. $$v\sim w$$ if there is $$u\in U$$ such that $$v=w+u$$. I know that we can prove $$\sim$$ is an equivalence relation on $$V$$ by showing it is reflexive, symmetric and transitive.

So far I have:

(reflexive) $$v = v+0$$ for all $$v\in V$$ so $$v\sim v$$ and it is transitive.

(symmetric) $$v=w+u, \quad u=(-w)+v$$ so $$w\sim v$$ and it is symmetric.

(transitive) $$v=u_{1}+w$$ and $$w=u_{2}+z$$

$$v+w=u_{1}+u_{2}+w+z$$

$$v=u_{1}+u_{2}+x$$

so $$v\sim z$$ and it is transitive. Am I doing it right?

• What are $V$ and $U$? – Arthur Oct 2 '19 at 11:50

Assuming that $$V$$ is a vector space, and $$U$$ is assumed to be a subspace of $$V$$, your proof has the right gist, but it's a bit sloppy. The first thing should be something like:

(reflexive) We need to show that for all $$v \in V$$, there's a $$u \in U$$ with $$v+u = v$$. Because $$U$$ is a subspace, and therefore contains the 0-vector, we can choose $$u = 0$$ and get $$v + u = v + 0 = v$$.

I grant you that's a bit wordy, but it does state explicitly what needs to be shown, and then explains why $$0$$ is allowed as a possible "u" in the proof.

For the third one, I'll get you started:

(transitive) Suppose that $$v \sim w$$ and $$w \sim x$$; we need to show that $$v \sim x$$, i.e., that there's a vector $$u \in U$$ with $$v + u = x$$. From the first assumption, we know there's a vector $$u_1 \in U$$ with $$v + u_1 = w$$. From the second...

Of course, both of these are proofs suitable for someone who's just learning linear algebra. In a research paper, it would probably suffice to say something like "Because $$U$$ is a subspace of $$V$$, the relation $$v \sim w$$ iff $$v-w \in U$$ is evidently an equivalence relation." So there is (as usual in mathematics) a question of audience for any given proof.

Suppose $$u \sim v$$ and $$v \sim w$$. Then, $$u=v+x$$ and $$v=w+y$$ for some vectors $$x$$ and $$y$$ on $$\textsf U$$. Thus $$u=(w+y)+x=w+(y+x)$$ so, $$u \sim w$$ since $$y+x\in \textsf U$$ (assuming $$\textsf U$$ is a vector subspace of $$\textsf V$$).

Symmetry and transitivity are not correct.

You need that $$U$$ is a subspace of $$V$$.

(1) $$v\sim v$$, since $$v=v+0$$ with $$0\in U$$.

(2) If $$v\sim w$$, i.e., $$v=w+u$$ for some $$u\in U$$, then $$w = v+(-u)$$ with $$-u\in U$$, i.e., $$w\sim v$$.

(3) If $$v\sim w$$ and $$w\sim x$$, i.e., $$v=w+u$$ and $$w=x+u'$$ for some $$u,u'\in U$$, then $$x= w - u' = v - u - u' = v+(-u-u')$$ with $$-u-u'\in U$$, i.e., $$v\sim x$$.